539.3 Timoshenko-type asymptotic theory for thin multi-layered plates shells

Dimitrienko Y. I. (Bauman Moscow State Technical University), Yurin Y. V. (Bauman Moscow State Technical University)

THIN MULTI-LAYERED SHELLS, ASYMPTOTIC THEORY, ASYMPTOTIC EXPANSIONS, TIMOSHENKOTYPE THEORY, KIRCHHOFF — LOVE-TYPE THEORY


doi: 10.18698/2309-3684-2018-1-1640


The paper presents a new modification of asymptotic theory describing thin multi-layered shells with finite shear rigidity. It is based on asymptotic analysis of general threedimensional equations from the elasticity theory for multi-layered bodies. This modification allows us to derive averaged equations from a Timoshenko-type plate theory. We identified the small geometrical parameter and used it to carry out our asymptotic analysis. We stated local elasticity theory problems which may be solved analytically. We show that when only the dominant terms of asymptotic expansions are taken into account, an asymptotic theory will result in the averaged plate equations of the Kirchhoff — Love type. When taking into account those terms that follow the dominant ones in asymptotic series in a self-similar way as compared to the previous approximation, an asymptotic theory will lead to Timoshenko-type averaged equations. At the same time, theoretical accuracy of the resulting truncated asymptotic solution is as high as that of the solution according to a Kirchhoff — Love type theory. The asymptotic theory modification that we developed makes it possible to use explicit analytical expressions to compute all six stress tensor components for a multi-layered plate with a high degree of accuracy. We used our method to perform a numerical simulation of stresses and displacements in a multi-layered plate subjected to uniform pressure that causes the plate to bend. Numerical computations show that our Timoshenko-type asymptotic theory provides a similarly high accuracy of computing flexural, shear and lateral stresses as compared to a three-dimensional finite element solution over a very fine mesh and a Kirchhoff — Love-type asymptotic theory. A Timoshenko-type theory will provide a better result for computing buckling than a Kirchhoff — Love-type theory, especially for relatively short plates. When the displacement is longitudinal, a Timoshenko-type theory will only provide a good result for elongated plates.


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Димитриенко Ю.И., Юрин Ю.В. Асимптотическая теория типа Тимошенко для тонких многослойных пластин. Математическое моделирование и численные методы, 2018, № 1, с. 16-40



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